Comparison between solutions of the general dynamic equation and the kinetic equation for nucleation and droplet growth

Abstract

A comparison is made between two models of homogeneous nucleation and droplet growth. The first is a kinetic model yielding the master equations for the concentrations of molecular clusters. Such a model does not make an explicit distinction between nucleation and droplet growth. The second model treats nucleation and growth separately, fully ignoring stochastic effects, and leads to the continuous general dynamic equation (GDE). Problems in applying the GDE model are discussed. A numerical solution of the kinetic equation is compared with an analytic solution of the GDE for two different cases: (1) the onset of nucleation and (2) the nucleation pulse. The kinetic model yields the thickness of the condensation front in size space as a function of supersaturation and dimensionless surface tension. If the GDE is applied properly, solutions of the GDE and the kinetic equation agree, with the exception of very small clusters, near-critical clusters, and the condensation front.

Molecular-resolution simulations of new particle formation: Evaluation of common assumptions made in describing nucleation in aerosol dynamics models

“…It needs to be emphasized, however, that the initial formation rate, generally referred to as the nucleation rate, is not a property of some specific critical cluster size, but a quantity defined for all sizes starting from the initial agglomerate of two molecules, regardless of whether there exists a critical cluster or not. In cases where there exists an unambiguous critical size, the physical nature of the clustering process does not undergo any fundamental change at this size (Holten and van Dongen 2009). …”

“…The discrete GDE provides the physically accurate description of the dynamics of small particles Korhonen et al 2003;Holten and van Dongen 2009;Olenius et al 2015), but its applications are restricted to relatively small model systems, since the number of kinetic equations becomes computationally unbearable as the modeled size range increases to particles consisting of thousands of molecules and/or multiple chemical compounds. It is used in this work for the whole simulated size range in order to study the interface between the distribution below and above the assumed formation size without the uncertainties related to the continuous GDE and discretization errors (Korhonen et al 2003).…”

“…Models providing the formation rate typically treat the particles-or molecular clusters-below a couple of nanometers with more accurate, discrete molecular-resolution approaches (see, e.g., McGrath et al 2012, and references therein). These models explicitly describe the population dynamics by a set of discrete kinetic equations which accurately treat the molecular collisions and evaporations forming and growing the population Holten and van Dongen 2009).…”

“…Yet, there is no fundamental difference in the physics below and above the "formation size." The reason for separating the process into two different stages is primarily mathematical: the continuous GDE does not consider the stochastic widening of the particle size distribution, which plays a major role in the dynamics of newly formed particle populations at the smallest sizes of a couple of nanometers (Holten and van Dongen 2009;Wang et al 2013;Olenius et al 2015).…”

Comment on “Comparison between solutions of the general dynamic equation and the kinetic equation for nucleation and droplet growth” [J. Chem. Phys. 130, 014102 (2009)]

“…An interesting suggested way to avoid ambiguity in selecting the location of a stationary source is to request that the resulting sharp front give asymptotically the proper number of nuclei. 1 In terms of Eq. ͑2͒ this corresponds to x = ␥, the Euler constant, quite similarly to the time-lag ͑"induction time"͒ problem.…”